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  1. —Three linear (not coupled) O.D.E.'s for the body position in the inertial frame of reference whose solution yields the position of the body in the inertial frame of reference.

The above 13 vehicle dynamics O.D.E.'s are numerically integrated using a fourth-order accurate Runge-Kutta technique. The assembly and solution of the vehicle dynamics equations was implemented directly from the vehicle simulation code TRJv [ 32].

The coupled Navier-Stokes flow solver and the 6-DOF equations of motion solver was successfully used to simulate a free falling sphere (Re=1000) and a 6:1 ellipsoid (Re=7500) prior to the SUBOFF calculation presented later in this paper.


Turbulence models used in this work include the Baldwin-Lomax mixing length model [6] the Launder-Sharma Low-Reynolds number k–ε model and a Low-Reynolds number nonlinear k–ε model based on the work of Nisizima and Yoshizawa, Speziale, and Myong and Kasagi [7].

The bulk of the initial work was done with the algebraic mixing length model typified by the eddy viscosity

Here, a representative length scale is determined empirically, and it is combined with vorticity, ω, to provide a representative velocity scale.

A more general model uses two transport equations to find the representative length and velocity scales of the flow. In this work, the Launder-Sharma model was used to find the turbulent kinetic energy, k, and the turbulent energy dissipation rate, ε. The resulting velocity scale is k1/2 while the length scale is provided by k3/2/ε, so that the turbulent viscosity is given by

Damping functions, including one for the eddy viscosity, are used to allow integration of the flow directly to the wall. Both of the above models rely on the Boussinesq approximation, which provides for an isotropic eddy viscosity and so fails to capture important anisotropic effects. For this reason, the k–ε model was extended by a nonlinear Reynolds stress assumption of the form

where Sij and Ωij represent the mean strain and rotation tensors, respectively, and k and ε continue to be determined by the Launder-Sharma Low-Reynolds number k–ε model. The standard model is recovered when C1 and C2 are set to zero. However, the nonlinear effects provided by this new Reynolds stress assumption prove to be dramatic. Calculation in a three-dimensional square duct, shown in Fig.1, demonstrates secondary flow which is absent when a standard two-equation model is used. Additionally, it has been shown that the model greatly improves the prediction of the normal Reynolds stresses in highly three-dimensional flows, such as the flow over a 6:1 prolate spheroid at 10° angle of attack. Results, compared to the experimental data of Reference [33], are shown in Figs.25.

Fig. 1 W/Ur Profiles in the Square Duct.

Fig. 2 Boundary Layer Velocity Profile, x/L=0.4, Φ=100°, Anisotropic Model, Re=4.2×106, α=10°.

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